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Do you agree we can have branches (or histories) in space
(in a space) but also branches (or histories) in time?

I guess there is an implicit "not only" in this question :)

You have an atom, excited (ie by a laser).
This atom can radiate a photon in two different
ways (I mean two different transitions), having
the same life-time. Of course the photon has
the energy E1, or the (different) energy E2.
You can write the amplitude as ....
[a exp(-iE1 t/h) + b exp(-iE2 t/h)] e^(-k t)
The probability, for a transition to occur,
at a specific time, with emission of one photon
(it does not matter, here, if its energy is E1 or E2),
is given by two terms, plus an interference term.

Yes. The two terms in your equation correspond to the "micro branches" I
mentioned. Hence, I gather, you have demonstrated interference effects in
time. For those who have not seen this before, it means that you get
oscillations in the detection probability with time.

You can, of course, introduce decoherence in this sort
of 'interference in time' quantum experiment (quantum
beat), by filtering for energies, ie for E1 or for E2.
In that case the extra interference term disappears.

Yes, if by "filtering" you mean putting in physical filters so that one
photon is absorbed. The filter then provides the "environment" refered to
in the theory of decoherence.

To answer your initial question: interference effects are not branches.
Actually they imply the absence of effective branching.

You don't get branching in time because time is a parameter, not an
observable: this means that there is no quantum uncertainty about what the
time is. (At least in the non-relativistic theory. Frankly, I don't know
how to handle the relativistic case).

You might say: we don't know what time the particle will be detected. Yes,
but the theory doesn't consider the detection event as *one thing* with an
uncertain time. In the MWI there are many (a continuum) of detection
events, each of which happens at a well defined time and each of which
starts off its own branch. And the act of detection changes the detector
physically, which is to say that its particles are re-arranged. Hence the
slogan "every measurement is a position measurement". Of course they are
all momentum measurements as well, etc.